Advertisements
Advertisements
प्रश्न
Find the value of: x3 – 5x2 + 4x + 8, if x = `10/(3 - "i")`.
Advertisements
उत्तर
x = `10/(3 - "i")`
∴ x = `(10(3 + "i"))/((3 - "i")(3 + "i")`
= `(10(3 + "i"))/(9 - "i"^2)`
= `(10(3 + "i"))/(9 - (- 1)` ...[∵ i2 = –1]
= `(10(3 + "i"))/10`
∴ x = 3 + i
∴ x – 3 = i
∴ (x – 3)2 = i2
∴ x2 – 6x + 9 = –1 ...[∵ i2 = –1]
∴ x2 – 6x + 10 = 0 ...(i)
x + 1
`x^2 – 6x + 10")"overline(x^3 - 5x^2 + 4x + 8)"`
x3 – 6x2 + 10x
– + –
x2 – 6x + 8
x2 – 6x + 10
– + –
– 2
∴ x3 – 5x2 + 4x + 8
= (x2 – 6x + 10)(x + 1) – 2
= 0 (x + 1) – 2 ...[From (i)]
= 0 – 2
∴ x3 – 5x2 + 4x + 8 = – 2.
APPEARS IN
संबंधित प्रश्न
Find the value of i49 + i68 + i89 + i110
Write the conjugates of the following complex number:
5i
Show that 1 + i10 + i100 − i1000 = 0
Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
Answer the following:
Simplify the following and express in the form a + ib:
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
Solve the following equation for x, y ∈ R:
(4 − 5i)x + (2 + 3i)y = 10 − 7i
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
If z ≠ 1 and `"z"^2/("z - 1")` is real, then the point represented by the complex number z lies ______.
State true or false for the following:
The points representing the complex number z for which |z + 1| < |z − 1| lies in the interior of a circle.
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
Solve the equation |z| = z + 1 + 2i.
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
State True or False for the following:
The locus represented by |z – 1| = |z – i| is a line perpendicular to the join of (1, 0) and (0, 1).
If `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, then ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals to ______.
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9) / (i^6 + 2i^8 + 3i^18)`
